#511582

order	512
nilpotency class	4
isoclinism	isoclinic class of #511577
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

638 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,20)(19,25,31,32)(21,33,23,26)(22,27,39,34)(28,36,45,48)(29,43,40,41)(30,42,47,46)(35,37,38,44);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,24,31)(18,25,20,32)(21,29,23,40)(22,36,39,48)(26,41,33,43)(27,45,34,28)(30,35,47,38)(37,46,44,42);(1,3)(2,6)(4,8)(5,7)(10,14)(13,16)(17,20,24,18)(19,28,31,45)(21,33,23,26)(22,37,39,44)(25,36,32,48)(27,38,34,35)(29,46,40,42)(30,41,47,43);(3,8)(5,6)(11,15)(13,16)(17,21,24,23)(18,26,20,33)(19,29,31,40)(22,35,39,38)(25,41,32,43)(27,44,34,37)(28,46,45,42)(30,36,47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418778);